# Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

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### How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation?

I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-a\int^t_0 X_s\,ds +B_t$. B is the Brownian motion. And its analytic ...
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### Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$?

Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$ is independent on $a$?
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### Continuity and $L^p$ spaces

I have been wondering how to solve this question I saw in a textbook. Given $g \in \bigcup _{1\leq p\leq \infty} L^{p}$ define, for $r \in [ 0,1]$ , $$G(r) = \int_{0}^{r} g(t) dt \;.$$ Show that $G$...
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### Exponential integration rule

Given the function $e^ae^{-ba}$ The indefinite integral is $\int e^ae^{-ba} \mathrm da = \int e^{a-ba} \mathrm da$ is $\frac{e^ae^{-ba}}{1-b}$ I get that $\int e^u = e^u\frac{\mathrm du}{\mathrm da}$...
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### Integral over the full support of a square and cube of a convolution of normal and uniform

I've got a uniform random variable $X\sim\mathcal{U}(-a,a)$ and a normal random variable $Y\sim\mathcal{N}(0,\sigma^2)$. I am interested in their sum $Z=X+Y$. Using the convolution integral, one can ...
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### Numerical solution of the Dirichlet problem for unit circle in two dimensions

How to approach the problem using integral equations and solve it with numerical methods in matlab? Are there any tutorials with code for begginers?
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### Transformation of Random Variables and Expectation

Suppose we have $0\le\,X\le\,\infty$, and we have $Y = a+b\, X\,\quad$ where $a\in\mathbb{R}$, $b\in\mathbb{R}$, $X \sim \chi^2_\nu(\beta)\quad$ with a PDF $f_X(x)$. Now, I found that the PDF of Y is ...
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### tough integral involving the Cosine integral

I ran across an integral on a German math site that has a friend of mine and I quite stuck. They give, without derivation, \int_0^\infty \mathrm{Ci}(\alpha x)\mathrm{Ci}(\beta x)dx=\frac{\pi}{2 \...
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### Finding total mass of a plate using Integration

I am having difficulty parsing this question and visualizing how the picture looks and what tools to use to solve Question: A flat metal plate is in the shape determined by area by the area under the ...
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### What is the volume of this ring-like solid?

We're learning about triple integrals and such in class. Here's one of the problems I'm working on: A cylindrical drill with radius 3 is used to bore a hole through the center of a sphere of radius ... I am wondering ifmy curves look like $y=9-x^2, z=x^2-3x$ for area between curves, why isit just $\int^{3}_{-3/2}(9-x^2)-(x^2-3x) dx$ I don’t care if there’s a change of sign?